Traveling fronts on networks under reaction - diffusion equations

Joachim von Below
Université du Littoral

We consider reaction–diffusion processes on finite or locally finite networks governed by nonlinear autonomous reaction–diffusion equations on the edges and the continuity requirement
at ramification nodes. In addition, as a distinctive feature, we impose a possibly dynamical Kirchhoff flow conditions at the nodes vi of the form i@tu(vi, t) +Xj dijcij @xjuj(vi, t) = 0. It includes the classical weighted Kirchhoff flow incidence law corresponding to i = 0. The conditions cij > 0, i  0 lead to dissipativity, and a linear and semilinear global parabolic existence and regularity theory is well established. Under non dissipative Kirchhoff conditions blow up, excitation and nonuniqueness phenomena can occur. Anyhow, continuous
waves in networks obey canonically certain dynamical Kirchhoff conditions that, in general,
are not dissipative. Here we are interested in the occurrence of travelling front solutions, i.e. in similarity solutions of the parabolic model problem that are of wave form. It turns out that under the continuity requirement at ramification nodes only, the existence of a front continuum parametrized by a single branch speed, can be completely determined and characterized in
terms of the diffusion rates, the edge lengths and the graph orientation (dij)n×N
. Especially, up to inverting the orientation and up to translation, there is only one front corresponding to the respective minimal branch speeds. Hence, the Kirchhoff conditions can possibly be redundant, but also be impossible to satisfy. Conceivably, we impose these conditions only with constant coefficients and derive existence criteria in that case. Moreover, we discuss special flow laws as the classical consistent Kirchhoff law, the consistent dynamical one etc., and the occurrence of isotachic fronts under these conditions. Finally we discuss attractivity and stability properties of equilibria in the dissipative case.

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